1109.1212 (Bert Schroer)

The foundational origin of integrability in quantum field theory    [PDF]

Bert Schroer

There are two foundational model-independent ways of thinking about
integrability in QFT. One is "dynamical" and generalizes the solvability in
closed form known from the Kepler two-body problem and its quantum mechanical
counterpart. The other, referred to as kinematical integrability, has no
classical nor quantum mechanical counterpart; it describes the relation between
observable algebras and their inequivalent representation classes in form of a
discrete algebra describing the a group dual or a braid group representation.
The dynamical integrability is defined in terms of properties of wedge
localization and uses the fact that modular localization theory allows to
"emulate" the application of operators of the incoming wedge-localized
interaction-free algebra inside the corresponding interacting algebra.
Emulation can be viewed as a kind of generalization of the intrinsic (no
Lagrangian quantization) Wigner particle representation theory to interacting
systems. It also leads to a profound understanding of crossing in particle
physics and places it into sharp difference to the kind of crossing used in the
dual model and string theory. Discrete integrability has a particular
interesting realization in higher dimensional conformal theories where the
spectrum of anomalous scale dimensions turns out to be a special aspect of this
integrability. In its formulation the Huygens principle (timelike commutation)
of observables play an important role. Since QFT is the theory which arises
from the principle of causal localization, dynamical as well as discrete
integrability have their foundational origin in localization.

View original: http://arxiv.org/abs/1109.1212